Weak Convergence Of A Class Of Operator's Fixed Point Theorem And A Class Of Image Theorem

The method of iterative approximation is one of the basic tools to deal with nonlinear problems, particular to the nonlinear operators which satisfy appropriate ordered conditions. In this paper, the first major work is based on this theory. We prove some fixed point theorems of variable order operators and mixed monotone mappings while having ordered conditions. The results obtained in this paper improve and extend the corresponding results of fixed point of ordered contractive maps. The existence and uniqueness of fixed point are also discussed in this paper.The second major work is based on the development and improvement of iterative theory. In the third section, the modified Mann algorithm in 2-uniformly smooth Banach spaces generates a sequence {x_n} by the formula:Where C is a closed convex subset of a 2-uniformly smooth Banach space, T:Câ†'C is aκ-strict pseudo-contraction. It is proved that if the control sequence {α_n} is chosen to satisfy appropriate conditions, then {x_n} convergesweakly to a fixed point of T.In the forth section, weak convergence theorems for asymptoticallyκ-strict pseudo-contraction in Hilbert space are proved. The sequence {x_n} is generated by the following algorithms :Where C is a closed convex subset of a Hilbert space H,T :Câ†'H is a asymptoticallyκ-strict pseudo-contraction, S:Câ†'H is defined by Sx=κx+(1-κ)T~nx,Pc is metric projection H onto C. It is proved that if {α_n} and {β_n} are chosen to satisfy appropriate conditions, then {x_n} converges weakly to a fixed point of T.